About the Project

Next 10 matching pages

basic hypergeometric functions

♦ 1—10 of 36 matching pages ♦

(0.004 seconds)

1—10 of 36 matching pages

1: Howard S. Cohl

… ►His research interests include fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre and Jacobi functions, generalized and basic hypergeometric functions, eigenfunction expansions in separable coordinate systems, generating functions,

q

-series, and orthogonal polynomials in the Askey and

q

-Askey schemes. …

2: 17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …

3: 17.1 Special Notation

§17.1 Special Notation

… ►The main functions treated in this chapter are the basic hypergeometric (or

q

-hypergeometric) function

{{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, the bilateral basic hypergeometric (or bilateral

q

-hypergeometric) function

{{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, and the

q

-analogs of the Appell functions

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)

,

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)

,

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)

, and

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)

. …

4: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

… ►

17.4.1

{{}_{r+1}\phi_{s}}\left({a_{0},a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,% b_{s}};q,z\right)={{}_{r+1}\phi_{s}}\left(a_{0},a_{1},\dots,a_{r};b_{1},b_{2},% \dots,b_{s};q,z\right)=\sum_{n=0}^{\infty}\frac{\left(a_{0};q\right)_{n}\left(% a_{1};q\right)_{n}\cdots\left(a_{r};q\right)_{n}}{\left(q;q\right)_{n}\left(b_% {1};q\right)_{n}\cdots\left(b_{s};q\right)_{n}}\*\left((-1)^{n}q^{\genfrac{(}{% )}{0.0pt}{}{n}{2}}\right)^{s-r}z^{n}.

ⓘ

Defines:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §17.6, §18.27(i), §18.27(ii)
Permalink:: http://dlmf.nist.gov/17.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(i), §17.4 and Ch.17

… ►

17.4.2

\lim_{q\to 1-}{{}_{r+1}\phi_{s}}\left({q^{a_{0}},q^{a_{1}},\dots,q^{a_{r}}% \atop q^{b_{1}},\dots,q^{b_{s}}};q,(q-1)^{s-r}z\right)={{}_{r+1}F_{s}}\left({a% _{0},a_{1},\dots,a_{r}\atop b_{1},\dots,b_{s}};z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Source:: Koekoek et al. (2010, p. 15)
Referenced by:: Erratum (V1.1.11) for Equation (17.4.2)
Permalink:: http://dlmf.nist.gov/17.4.E2
Encodings:: pMML, png, TeX
Generalization (effective with 1.1.11):: This limit relation which was previously accurate for ${{}_{r+1}\phi_{r}}$ was updated to be accurate for the more general ${{}_{r+1}\phi_{s}}$ .
See also:: Annotations for §17.4(i), §17.4 and Ch.17

… ►

17.4.3

{{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.

ⓘ

Defines:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.8
Permalink:: http://dlmf.nist.gov/17.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(ii), §17.4 and Ch.17

… ►

17.4.4

\lim_{q\to 1-}{{}_{r}\psi_{r}}\left({q^{a_{1}},q^{a_{2}},\dots,q^{a_{r}}\atop q% ^{b_{1}},q^{b_{2}},\dots,q^{b_{r}}};q,z\right)={{}_{r}H_{r}}\left({a_{1},a_{2}% ,\dots,a_{r}\atop b_{1},b_{2},\dots,b_{r}};z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}H_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : bilateral hypergeometric function, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $q$ : complex base, $r$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(ii), §17.4 and Ch.17

…

5: 17.15 Generalizations

§17.15 Generalizations

►For higher-dimensional basic hypergometric functions, see Milne (1985a, b, c, d, 1988, 1994, 1997) and Gustafson (1987).

6: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

… ►

17.5.1

{{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty};

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)
Permalink:: http://dlmf.nist.gov/17.5.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.11):: The constraint $\left|z\right|<1$ is not necessary and was removed.
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.2

{{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},

\left|z\right|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Referenced by:: §17.5, Erratum (V1.2.1) for §17.5
Permalink:: http://dlmf.nist.gov/17.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.3

{{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.4

{{}_{1}\phi_{0}}\left(0;-;q,z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},

\left|z\right|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii), §17.5, Erratum (V1.2.1) for §17.5
Permalink:: http://dlmf.nist.gov/17.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.5

{{}_{1}\phi_{1}}\left({a\atop c};q,c/a\right)=\frac{\left(c/a;q\right)_{\infty% }}{\left(c;q\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function and $q$ : complex base
Referenced by:: Erratum (V1.1.10) for Equation (17.5.5)
Permalink:: http://dlmf.nist.gov/17.5.E5
Encodings:: pMML, png, TeX
Correction (effective with 1.1.10):: The constraint $\left|c\right|<\left|a\right|$ is not necessary and was removed.
See also:: Annotations for §17.5, §17.5 and Ch.17

7: Bibliography

… ►

G. E. Andrews (1966a) On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

ⓘ

External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §17.6(ii).
Encodings:: BibTeX

… ►

G. E. Andrews (1974) Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

ⓘ

External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §17.16, Ch.17.
Encodings:: BibTeX

…

8: 17.6 ${{}_{2}\phi_{1}}$ Function

… ►

17.6.2

{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,\ifrac{cq^{n}}{a}\right)=\frac{\left% (c/a;q\right)_{n}}{\left(c;q\right)_{n}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Referenced by:: §17.6(i)
Permalink:: http://dlmf.nist.gov/17.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.3

{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,q\right)=\frac{a^{n}\left(c/a;q% \right)_{n}}{\left(c;q\right)_{n}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.17

{{}_{2}\phi_{1}}\left({a,b\atop c/q};q,z\right)-{{}_{2}\phi_{1}}\left({a,b% \atop c};q,z\right)=cz\frac{(1-a)(1-b)}{(q-c)(1-c)}{{}_{2}\phi_{1}}\left({aq,% bq\atop cq};q,z\right),

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

►

17.6.18

{{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-{{}_{2}\phi_{1}}\left({a,b\atop c% };q,z\right)=az\frac{1-b}{1-c}{{}_{2}\phi_{1}}\left({aq,bq\atop cq};q,z\right),

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

… ►

17.6.21

b(1-a){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-a(1-b){{}_{2}\phi_{1}}% \left({a,bq\atop c};q,z\right)=(b-a){{}_{2}\phi_{1}}\left({a,b\atop c};q,z% \right),

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

…

9: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

17.9.1

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(za;q\right)_{\infty}% }{\left(z;q\right)_{\infty}}{{}_{2}\phi_{2}}\left({a,c/b\atop c,az};q,bz\right),

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

… ►

17.9.5

{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{n% }}{\left(c;q\right)_{n}}{{}_{3}\phi_{2}}\left({q^{-n},b,bzq^{-n}/c\atop bq^{1-% n}/c,0};q,q\right).

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

… ►

17.9.8

{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(de/(bc);q% \right)_{n}}{\left(e;q\right)_{n}}\left(\frac{bc}{d}\right)^{n}{{}_{3}\phi_{2}% }\left({q^{-n},d/b,d/c\atop d,de/(bc)};q,q\right),

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

►

17.9.9

{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(e/c;q\right% )_{n}}{\left(e;q\right)_{n}}c^{n}{{}_{3}\phi_{2}}\left({q^{-n},c,d/b\atop d,cq% ^{1-n}/e};q,\frac{bq}{e}\right),

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

►

17.9.10

{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,\frac{deq^{n}}{bc}\right)=\frac{% \left(e/c;q\right)_{n}}{\left(e;q\right)_{n}}{{}_{3}\phi_{2}}\left({q^{-n},c,d% /b\atop d,cq^{1-n}/e};q,q\right).

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

…

10: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

… ►

17.10.1

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.2

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.3

{{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.5

\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.6

\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \*{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{3}}{bcdefghk}\right)=\frac{\left(q,q% /(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q\right)_{\infty}}{% \left(gh,gk,h/g,k/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{\infty}}\*{{}_{10}\phi% _{9}}\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg% /f,qg/h,qg/k};q,\frac{q^{3}}{bcdefghk}\right)+\operatorname{idem}\left(g;h,k% \right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $k$ : nonnegative integer and $q$ : complex base
Referenced by:: Erratum (V1.2.4) for Equation (17.10.6)
Permalink:: http://dlmf.nist.gov/17.10.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.2.4):: In the numerator of the argument of the basic bilateral hypergeometricfunction ${{}_{10}\psi_{10}}$ and in the numerator of the arguments of the basichypergeometric ${{}_{10}\phi_{9}}$ functions, we replaced $q^{2}$ by $q^{3}$ . We also added a missing factor $1/\left(k/g;q\right)_{\infty}$ in the first term on the right-hand side.
See also:: Annotations for §17.10, §17.10 and Ch.17

Next 10 matching pages

© 2010–2026 NIST / Disclaimer / Feedback.

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov